# American Institue of Mathematical Sciences

## Pullback attractors for a class of non-autonomous thermoelastic plate systems

 1 Universidade Federal da Paraíba Departamento de Matemática 58051-900 João Pessoa PB, Brazil 2 Universidade Federal de São Carlos, Departamento de Matemática, 13565-905 São Carlos SP, Brazil

* Corresponding author

The first author is partially supported by FAPESP grant #2014/03686-3, Brazil.
The third author is partially supported by FAPESP grant #2014/03109-6, Brazil.

Received  April 2017 Revised  July 2017 Published  September 2017

In this article we study the asymptotic behavior of solutions, in the sense of pullback attractors, of the evolution system
 $\begin{cases}u_{tt} +Δ^2 u+a(t)Δθ=f(t,u),&t>τ,\ x∈Ω,\\θ_t-κΔ θ-a(t)Δ u_t=0,&t>τ,\ x∈Ω,\end{cases}$
subject to boundary conditions
 $u=Δ u=θ=0,\ t>τ,\ x∈\partial Ω,$
where $Ω$ is a bounded domain in $\mathbb{R}^N$ with $N≥q 2$, which boundary $\partialΩ$ is assumed to be a $\mathcal{C}^4$-hypersurface, $κ>0$ is constant, $a$ is an Hölder continuous function and $f$ is a dissipative nonlinearity locally Lipschitz in the second variable. Using the theory of uniform sectorial operators, in the sense of P. Sobolevskiǐ ([23]), we give a partial description of the fractional power spaces scale for the thermoelastic plate operator and we show the local and global well-posedness of this non-autonomous problem. Furthermore we prove existence and uniform boundedness of pullback attractors.
Citation: Flank D. M. Bezerra, Vera L. Carbone, Marcelo J. D. Nascimento, Karina Schiabel. Pullback attractors for a class of non-autonomous thermoelastic plate systems. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2017214
##### References:
 [1] H. Amann, Linear and Quasilinear Parabolic Problems. Volume Ⅰ: Abstract Linear Theory Birkhäuser Verlag, Basel, 1995. [2] D. Andrade, M. A. Jorge Silva, T. F. Ma, Exponential stability for a plate equation with p-Laplacian and memory terms, Math. Meth. Appl. Sci., 35 (2012), 417-426. doi: 10.1002/mma.1552. [3] R. O. Araújo, To Fu Ma, Y. Qin, Long-time behavior of a quasilinear viscoelastic equation with past history, J. Differential Equations, 254 (2013), 4066-4087. doi: 10.1016/j.jde.2013.02.010. [4] A. R. A. Barbosa, T. F. Ma, Long-time dynamics of an extensible plate equation with thermal memory, J. Math. Anal. Appl., 416 (2014), 143-165. doi: 10.1016/j.jmaa.2014.02.042. [5] M. Baroun, S. Boulite, T. Diagana, L. Maniar, Almost periodic solutions to some semilinear non- autonomous thermoelastic plate equations, J. Math. Anal. Appl., 349 (2009), 74-84. doi: 10.1016/j.jmaa.2008.08.034. [6] M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. Ferreira, Existence and uniform decay for a non-linear viscoelastic equation with strong damping, Math. Methods Appl. Sci., 24 (2001), 1043-1053. doi: 10.1002/mma.250. [7] T. Caraballo, A. N. Carvalho, J. A. Langa, F. Rivero, A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor, Nonlinear Anal., 74 (2011), 2272-2283. doi: 10.1016/j.na.2010.11.032. [8] T. Caraballo, A. N. Carvalho, J. A. Langa, F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes, Nonlinear Anal., 72 (2010), 1967-1976. doi: 10.1016/j.na.2009.09.037. [9] T. Caraballo, G. Lukaszewicz, J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Analysis, 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111. [10] V. L. Carbone, M. J. D. Nascimento, K. Schiabel-Silva, R. P. Silva, Pullback attractors for a singularly nonautonomous plate equation, Electron. J. Differential Equations, 77 (2011), 1-13. [11] A. N. Carvalho, J. W. Cholewa, Local well-posedness for strongly damped wave equations with critical nonlinearities, Bull. Austral. Math. Soc., 66 (2002), 443-463. doi: 10.1017/S0004972700040296. [12] A. N. Carvalho, J. W. Cholewa, T. Dlotko, Damped wave equations with fast growing dissipative nonlinearities, Discrete Contin. Dyn. Syst., 24 (2009), 1147-1165. doi: 10.3934/dcds.2009.24.1147. [13] A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Non-autonomous Dynamical Systems Applied Mathematical Sciences, 182, Springer-Verlag, 2013. [14] A. N. Carvalho, M. J. D. Nascimento, Singularly non-autonomous semilinear parabolic problems with critical exponents and applications, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 449-471. [15] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics AMS Colloquium Publications v. 49, A. M. S, Providence, 2002. [16] C. Giorgi, J. E. Muñoz Rivera, V. Pata, Global attractors for a semilinear hyperbolic equation in viscoelasticity, J. Math. Anal. Appl., 260 (2001), 83-99. doi: 10.1006/jmaa.2001.7437. [17] D. Henry, Geometric Theory of Semilinear Parabolic Equations Lecture Notes in Mathematics 840, Springer-Verlag, Berlin, 1981. [18] I. Lasiecka, R. Triggiani, Analyticity of thermo-elastic semigroups with free boundary conditions, Ann. Sc. Norm. Super. Pisa Cl. Sci., 27 (1998), 457-482. [19] I. Lasiecka, R. Triggiani, Analyticity, and lack thereof, on thermo-elastic semigroups, Contrȏle et Équations aux Dérivées Partielles, ESAIM: Proceedings, 4 (1998), 199-222. [20] Z. Y. Liu, M. Renardy, A note on the equations of a thermoelastic plate, Appl. Math. Lett., 8 (1995), 1-6. doi: 10.1016/0893-9659(95)00020-Q. [21] Z. Liu, S. Zheng, xponential stability of the Kirchhoff plate with thermal or viscoelastic damping, Quart. Appl. Math., 55 (1997), 551-564. doi: 10.1090/qam/1466148. [22] Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems In CRC Research Notes in Mathematics 398, Chapman and Hall, 1999. [23] P. E. Sobolevskiǐ, Equations of parabolic type in a Banach space, Amer. Math. Soc. Transl., 49 (1966), 1-62. [24] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators Veb Deutscher, 1978.

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##### References:
 [1] H. Amann, Linear and Quasilinear Parabolic Problems. Volume Ⅰ: Abstract Linear Theory Birkhäuser Verlag, Basel, 1995. [2] D. Andrade, M. A. Jorge Silva, T. F. Ma, Exponential stability for a plate equation with p-Laplacian and memory terms, Math. Meth. Appl. Sci., 35 (2012), 417-426. doi: 10.1002/mma.1552. [3] R. O. Araújo, To Fu Ma, Y. Qin, Long-time behavior of a quasilinear viscoelastic equation with past history, J. Differential Equations, 254 (2013), 4066-4087. doi: 10.1016/j.jde.2013.02.010. [4] A. R. A. Barbosa, T. F. Ma, Long-time dynamics of an extensible plate equation with thermal memory, J. Math. Anal. Appl., 416 (2014), 143-165. doi: 10.1016/j.jmaa.2014.02.042. [5] M. Baroun, S. Boulite, T. Diagana, L. Maniar, Almost periodic solutions to some semilinear non- autonomous thermoelastic plate equations, J. Math. Anal. Appl., 349 (2009), 74-84. doi: 10.1016/j.jmaa.2008.08.034. [6] M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. Ferreira, Existence and uniform decay for a non-linear viscoelastic equation with strong damping, Math. Methods Appl. Sci., 24 (2001), 1043-1053. doi: 10.1002/mma.250. [7] T. Caraballo, A. N. Carvalho, J. A. Langa, F. Rivero, A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor, Nonlinear Anal., 74 (2011), 2272-2283. doi: 10.1016/j.na.2010.11.032. [8] T. Caraballo, A. N. Carvalho, J. A. Langa, F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes, Nonlinear Anal., 72 (2010), 1967-1976. doi: 10.1016/j.na.2009.09.037. [9] T. Caraballo, G. Lukaszewicz, J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Analysis, 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111. [10] V. L. Carbone, M. J. D. Nascimento, K. Schiabel-Silva, R. P. Silva, Pullback attractors for a singularly nonautonomous plate equation, Electron. J. Differential Equations, 77 (2011), 1-13. [11] A. N. Carvalho, J. W. Cholewa, Local well-posedness for strongly damped wave equations with critical nonlinearities, Bull. Austral. Math. Soc., 66 (2002), 443-463. doi: 10.1017/S0004972700040296. [12] A. N. Carvalho, J. W. Cholewa, T. Dlotko, Damped wave equations with fast growing dissipative nonlinearities, Discrete Contin. Dyn. Syst., 24 (2009), 1147-1165. doi: 10.3934/dcds.2009.24.1147. [13] A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Non-autonomous Dynamical Systems Applied Mathematical Sciences, 182, Springer-Verlag, 2013. [14] A. N. Carvalho, M. J. D. Nascimento, Singularly non-autonomous semilinear parabolic problems with critical exponents and applications, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 449-471. [15] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics AMS Colloquium Publications v. 49, A. M. S, Providence, 2002. [16] C. Giorgi, J. E. Muñoz Rivera, V. Pata, Global attractors for a semilinear hyperbolic equation in viscoelasticity, J. Math. Anal. Appl., 260 (2001), 83-99. doi: 10.1006/jmaa.2001.7437. [17] D. Henry, Geometric Theory of Semilinear Parabolic Equations Lecture Notes in Mathematics 840, Springer-Verlag, Berlin, 1981. [18] I. Lasiecka, R. Triggiani, Analyticity of thermo-elastic semigroups with free boundary conditions, Ann. Sc. Norm. Super. Pisa Cl. Sci., 27 (1998), 457-482. [19] I. Lasiecka, R. Triggiani, Analyticity, and lack thereof, on thermo-elastic semigroups, Contrȏle et Équations aux Dérivées Partielles, ESAIM: Proceedings, 4 (1998), 199-222. [20] Z. Y. Liu, M. Renardy, A note on the equations of a thermoelastic plate, Appl. Math. Lett., 8 (1995), 1-6. doi: 10.1016/0893-9659(95)00020-Q. [21] Z. Liu, S. Zheng, xponential stability of the Kirchhoff plate with thermal or viscoelastic damping, Quart. Appl. Math., 55 (1997), 551-564. doi: 10.1090/qam/1466148. [22] Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems In CRC Research Notes in Mathematics 398, Chapman and Hall, 1999. [23] P. E. Sobolevskiǐ, Equations of parabolic type in a Banach space, Amer. Math. Soc. Transl., 49 (1966), 1-62. [24] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators Veb Deutscher, 1978.
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